Triangulation Supplemental

1
Triangulation Supplemental

• From ORourke (Chs. 12)
• Fall 2005

2
Contents

• Ear clipping algorithm
• Triangulating monotonic polygons
• Monotonic decomposition via trapezoidalization

3
Ear Clipping

• Go around the polygon to check whether a diagonal
  can be drawn from (i-1) to (i1)
• Diagonal check
• No edge crossing AND inside P

What is the time complexity in the worst case?
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Think

• Can every simple polygon be triangulated?
• Does every simple polygon have at two ears (so
  that ear clipping algorithm can work)?

6
Validity of Diagonal

• No edge crossing
• Each potential diagonal (i-1, i1) need to check
  with ? edges

• In-Cone check
• local geometry

7
Triangulating Monotonic Polygons
Why?

• Linear time algorithm O(n)
• Fact on monotonic polygons
• Def a vertex is called reflex if its internal
  angle is strictly greater than p
• Def cusp
• A reflex vertex whose adjacent vertices v- and v
  are either both above or below v.
• Lemma
• If a polygon P has no cusps, then it is monotone.

8
Algorithm Ideas

• Cut off triangles from the top in a greedy
  fashion
• At each step, the first available triangle
  removed
• For each vertex v, connect v to all the vertices
  above it and visible via a diagonal, and remove
  the top portion of the polygon thereby
  triangulated
• Continue with the next vertex below v

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Case 1
Case 2a, 2b
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Triangulate Monotone Polygon
Sort by y-coordinate reflex chain 1,2 v 3
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v 3 chain 1,2 same side, non-convex
Case 2b chain 1,2,3 v 4
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v 4 chain 1,2,3 same side, non-convex
Case 2b chain 1,2,3,4 v 5
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v 5 chain 1,2,3,4 opposite side
Case 1 diagonal (5,2) reflex chain 1,2,3,4
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v 5 chain 2,3,4 opposite side
Case 1 diagonal (5,3) reflex chain 2,3,4
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v 5 chain 3,4 opposite side
Case 1 diagonal (5,4) reflex chain 3,4
reflex chain 4,5 v 6
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v 6 chain 4,5 same side, non-convex
Case 2b reflex chain 4,5,6 v 7
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v 7 chain 4,5,6 same side, convex
Case 2a diagonal (7,5) reflex chain 4,5,6
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v 7 chain 4,5 same side, convex
Case 2a diagonal (7,4) reflex chain 4,5
reflex chain 4,7 v 8
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v 8 chain 4,7 lowest vertex stop
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0,1,2
0,1,2,3
0,1,2,3,4
0,1,2,3,4
4,5
4,6
6,7
7,8
7,9
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Trapezoidalization

• Accomplish monotonic subdivision via horizontal
  trapezoidalization
• Supporting vertices the vertices through which
  the horizontal lines are drawn
• Assume P be a polygon with no two vertices on a
  horizontal line
• Each trapezoid has exactly two supporting
  vertices one on top, one on bottom
• Remove cusps by connecting the supporting vertex
  to the opposite vertex

Assume no two points have same y coord.
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The support line stops at the boundary
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Trapezoidalization via Plane Sweep

• Time complexity O(n log n)
• Maintain a balanced tree of edges

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Sweep Line Events
(, a, c, b, )
(, a, d, b, )
(, a, c, d, b, )
(, a, b, )
(, a, b, )
(, a, c, d, b, )
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Summary
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Review Questions

• Analyze the worst case complexity of ear clipping
  algorithm
• Analyze the time complexity of the algorithm for
  triangulating monotone polygon
• Triangulate (and keep track of the reflex chain)
  of the polygon on the next page
• Decompose the polygon on page 24 into monotonic
  pieces using trapezoidalization

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Monotone-ization

• If the polygon has no cusp, it is monotone and
  can be triangulated immediately.
• seek an orientation where the polygon is
  monotone
• Decompose into monotone pieces via horizontal
  trapezoidalization

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Three Types of Event Points
Regular
(, a, c, b, )
replace
(, a, d, b, )
Upward cusp
remove
(, a, c, d, b, )
(, a, b, )
Downward cusp
insert
(, a, b, )
(, a, c, d, b, )
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Making Trapezoids

• Regular point Extend toward material side until
  a boundary is reached
• Cusps extend both sides until boundaries are
  reached

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aj
j
ajih
h
ajicdhu.cusp
a
i
ajicdg
acdgd.cusp
acdefgu.cusp
d
g
bcdefg
f
e
bcde
c
bc
b

Upward cusp connect to support vertex
above Downward cusp connect to support vertex
below implementation